Higher-dimensional order theory

The word “set” has two uses in mathematics:

1. in a sentence like “a group is a set equipped with the following operations (…), such that the following axioms are satisfied (…)”, the word “set” denotes a base domain for mathematical structures, equipped with an equality relation (needed to express the axioms of group);
2. in the phrase “the set of even numbers”, the word “set” denotes a subdomain of the domain of natural numbers (which is a set in the first meaning).

One dogma of 20th century mathematics was that one cannot define sets. But this applies only to sets in the sense of objects of a so-called set theory, like ZF, not to sets in the two mathematical meanings, which can be defined.

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The goal of this blog is mainly to advertise two nonstandard points of view on category theory (and, in particular, mathematics) that I’ve adopted 6 years ago and which gave me a clearer picture of category theory:

1. one should define categories without an equality at the level of objects (so that equality is defined only between arrows between two fixed objects);
2. the right higher-dimensional hierarchy is not the hierarchy of n-categories (sets, categories, 2-categories, …) but that of n-orders (1, truth values, orders, 2-orders (Ord-enriched categories), 3-orders, …).

The first point has been defended, among others, by constructivists (for example Roger Apéry) and by Michael Makkai. It is also usually the case in the formalisations of category theory in proof assistants that the definition of category does not include an equality relation on objects.

The n-order hierarchy is the natural extension (by iterated enrichment) of the (-2)- and (-1)-categories (i.e. 1 and the truth values) discovered by Toby Bartels and James Dolan and also follows from the first point: for example, the right notion of “1-category of dimension 0” cannot be a category whose arrows are all identities (i.e. sets, in the traditional way), since one can compare arrows only between two fixed objects, but rather a category where all arrows between fixed objects are equal (i.e. orders). Since that time, this hierarchy appeared in the Lectures on n-Categories and Cohomology by Baez and Shulman and on the nLab, but surprisingly it seems not to have been adopted by working categorists (and mathematicians).

The second point has given its (current) name to the blog: Higher-dimensional order theory, which I consider to be a better name for what is usually called category theory.

(pdf file of this entry)